The result is (b− a) y=mb (x − a) — m'a (x − b). This equation is of the first degree, and therefore the required locus is a straight line. (4) To find the centres of the inscribed circle and of the escribed circles of a triangle whose angular points are given. Let (x', y′), (x', y'), (x''', y''') be the angular points A, B, C respectively. The equation of BC is y (x"" - x')-x (y"" -y')+y""'x' -x"'y' =-0.(ii), and the equation of AB is y (x' - x') - x (y' — y')+y'x" — x'y"=0............... (iii). The perpendiculars on these lines from the centre of any one of the circles are equal in magnitude. The centres of the four circles are therefore [Art. 31] given by If the co-ordinates of the angular points A, B, C of the triangle be substituted in the equations (i), (ii), (iii) respectively, the left hand members of all three will be the same. Hence, [Art. 26] the angular points of the triangle are either all on the positive sides of the lines (i), (ii), (iii), or all on the negative sides. The perpendiculars from the centre of the inscribed circle on the sides of the triangle are all drawn in the same direction as those from the angular points of the triangle. Hence in (iv) the signs of all the ambiguities are positive for the inscribed circle. For the escribed circles the signs are respectively. ++, +, and ++ + EXAMPLES ON CHAPTER II. 1. A straight line moves so that the sum of the reciprocals of its intercepts on two fixed intersecting lines is constant; shew that it passes through a fixed point. 2. Prove that bx-2hxy+ay=0 represents two straight lines at right angles respectively to the straight lines ax2+2hxy + by2=0. 3. Find the equation to the n straight lines through (a, b) perpendicular respectively to the lines given by the equation 4. Find the angles between the straight lines represented by the equation ▾ 5. QA, OB are two fixed straight lines, A, B being fixed points; P, Q are any two points on these lines such that the ratio of AP to BQ is constant; shew that the locus of the middle point of PQ is a straight line. 6. If a straight line be such that the sum of the perpendiculars upon it from any number of fixed points is zero, shew that it will pass through a fixed point. 7. PM, PN are the perpendiculars from a point P on two fixed straight lines which meet in O; MQ, NQ are drawn parallel to the fixed straight lines to meet in Q; prove that, if the locus of P be a straight line, the locus of Q will also be a straight line. 8. A straight line OPQ is drawn through a fixed point 0, meeting two fixed straight lines in the points P, Q, and in the straight line OPQ a point R is taken such that OP, OR, OQ are in harmonic progression; shew that the locus of R is a straight line. 9. Find the equations of the diagonals of the parallelogram formed by the lines where and a' = c, a=0, a=c, a' =0, 10. ABCD is a parallelogram. Taking A as pole, and AB as initial line, find the polar equations of the four sides and of the two diagonals. 11. From a given point (h, k) perpendiculars are drawn to the axes and their feet are joined; prove that the length of the perpendicular from (h, k) upon this line is hk sinR w √{h2 + k2 + 2hk cos w}' and that its equation is hx – ky = l3 — k3. 12. The distance of a point (x,, y,) from each of two straight lines, which pass through the origin of co-ordinates, is 8; prove that the two lines are given by (x,y− xy,)2 = 8o (x2 + y2). 13. Shew that the lines FC, KB, and AL in the figure to Euclid 1. 47 meet in a point. 14. Find the equations of the sides of a square the co-ordinates of two opposite angular points of which are 3, 4 and 1, -1. 15. Find the equation of the locus of the vertex of a triangle which has a given base and given difference of base angles. 16. Find the equation of the locus of a point at which two given portions of the same straight line subtend equal angles. 17. The product of the perpendiculars drawn from a point on the lines is equal to the square of the perpendicular drawn from the same point on the line shew that the equation of the locus of the point is x2 + y2 = a3. • 18. PA, PB are straight lines passing through the fixed points A, B and intercepting a constant length on a given straight line; find the equation of the locus of P. 19. The area of the parallelogram formed by the straight, lines 3x+4y=7a, 3x+4y=7α, 4x+3y=761, and 4x+3y=7b ̧ is 7 (a, -a,) (b, — b2). 20. Shew that the area of the triangle formed by the lines ax* + 2hxy + by2 = 0 and lx+my+n= = 0 is n3 √(h3 — ab) am3 - 2hlm + bl* * 21. Shew that the angle between one of the lines given by ax2 + 2hxy + by2 = 0, and one of the lines ax2 + 2hxy + by3 + λ (x2 + y3) = 0, is equal to the angle between the other two lines of the system. 22. Find the condition that one of the lines ax2+2hxy+by* = 0, may coincide with one of the lines a'x2+2h'xy+b'y' = 0. 23. Find the condition that one of the lines ax2 + 2hxy + by3 = 0, may be perpendicular to one of the lines a'x2+2h'xy+b'y' = 0. 24. Shew that the point (1, 8) is the centre of the inscribed circle of the triangle the equations of whose sides are 4y+3x=0, 12y-5x = 0, y-15=0, respectively. 25. Shew that the co-ordinates of the centre of the circle inscribed in the triangle the co-ordinates of whose angular points are (1, 2), (2, 3) and (3, 1) are (8+ √10) and (16-10). Find also the centres of the escribed circles distinguishing the different cases. 26. If the axes be rectangular, prove that the equation (x2 — 3y2) x = my (y3 – 3x3) represents three straight lines through the origin making equal angles with one another. 27. Shew that the product of the perpendiculars from the point (x, y) on the lines ax2 + 2hxy + by3 = 0, is equal to ax+2hx'y+by 28. If P1, P, be the perpendiculars from (x, y) on the straight lines ax2 + 2hxy + by3 = 0, prove that 29. 2 (p12 + p ̧3) {(a − b)2 + 4h3} = 2 (a − b) (ax2 — by2) ' +4h (a + b) xy + 4h2 (x2 + y3). Shew that the locus of a point such that the product of the perpendiculars from it upon the three straight lines represented by ay3 + by2x + cyx2 + dx3 = 0 is constant and equal to k3 is ay3 + by3x+ cyx2 + dx3 − k3 √ (a − c)2 + (b − d)3 = 0. 30. Shew that the condition that two of the lines represented by the equation Ax+3Bxy+3Cxy + Dy3 = 0 may be at right angles is A2+3AC + 3BD + D2 = 0. 31. Shew that the equation a (x*+y*) — 4bxy (x2 — y3) + 6cx2y2 = 0 represents two pairs of straight lines at right angles, and that, if 2b2 = a3 + 3ac, the two pairs will coincide. 32. The necessary and sufficient condition that two of the lines represented by the equation ay* + bxy3 + cx3y2 + dx3y + ex1 = 0 should be at right angles is (b + d) (ad + be) + (e − a)2 (a + c + e) = 0. 33. Shew that the straight lines joining the origin to the points of intersection of the two curves and ax2+2hxy+by+2gx=0, a'x2+2h'xy + b'y' + 2g'x=0, will be at right angles to one another, if g'(a+b) = g(a'+b'). 34. Prove that, if the perpendiculars from the angular points of one triangle upon the sides of a second meet in a point, the perpendiculars from the angular points of the second on the sides of the first will also meet in a point. 35. If the angular points of a triangle lie on three fixed straight lines which meet in a point, and two of the sides pass through fixed points, then will the third side also pass through a fixed point. |